Solve for x 4(4x+2)^2-19=37

$4 {(4 x + 2)}^{2} - 19 = 37$

Step 1

Add $19$ to both sides of the equation.

$4 {(4 x + 2)}^{2} = 37 + 19$

Step 2

Add $37$ and $19$ .

$4 {(4 x + 2)}^{2} = 56$

Step 3

Divide each term in $4 {(4 x + 2)}^{2} = 56$ by $4$ and simplify.

Divide each term in $4 {(4 x + 2)}^{2} = 56$ by $4$ .

$\frac{4 {(4 x + 2)}^{2}}{4} = \frac{56}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 {(4 x + 2)}^{2}}{4} = \frac{56}{4}$

Divide ${(4 x + 2)}^{2}$ by $1$ .

${(4 x + 2)}^{2} = \frac{56}{4}$

Simplify the right side.

Divide $56$ by $4$ .

${(4 x + 2)}^{2} = 14$

Step 4

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$4 x + 2 = \pm \sqrt{14}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$4 x + 2 = \sqrt{14}$

Subtract $2$ from both sides of the equation.

$4 x = \sqrt{14} - 2$

Divide each term in $4 x = \sqrt{14} - 2$ by $4$ and simplify.

Divide each term in $4 x = \sqrt{14} - 2$ by $4$ .

$\frac{4 x}{4} = \frac{\sqrt{14}}{4} + \frac{- 2}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{\sqrt{14}}{4} + \frac{- 2}{4}$

Divide $x$ by $1$ .

$x = \frac{\sqrt{14}}{4} + \frac{- 2}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

$x = \frac{\sqrt{14}}{4} + \frac{2 (- 1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{\sqrt{14}}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{\sqrt{14}}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{\sqrt{14}}{4} + \frac{- 1}{2}$

Move the negative in front of the fraction.

$x = \frac{\sqrt{14}}{4} - \frac{1}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$4 x + 2 = - \sqrt{14}$

Subtract $2$ from both sides of the equation.

$4 x = - \sqrt{14} - 2$

Divide each term in $4 x = - \sqrt{14} - 2$ by $4$ and simplify.

Divide each term in $4 x = - \sqrt{14} - 2$ by $4$ .

$\frac{4 x}{4} = \frac{- \sqrt{14}}{4} + \frac{- 2}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \sqrt{14}}{4} + \frac{- 2}{4}$

Divide $x$ by $1$ .

$x = \frac{- \sqrt{14}}{4} + \frac{- 2}{4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{\sqrt{14}}{4} + \frac{- 2}{4}$

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

$x = - \frac{\sqrt{14}}{4} + \frac{2 (- 1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = - \frac{\sqrt{14}}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Cancel the common factor.

$x = - \frac{\sqrt{14}}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Rewrite the expression.

$x = - \frac{\sqrt{14}}{4} + \frac{- 1}{2}$

Move the negative in front of the fraction.

$x = - \frac{\sqrt{14}}{4} - \frac{1}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{14}}{4} - \frac{1}{2}, - \frac{\sqrt{14}}{4} - \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{14}}{4} - \frac{1}{2}, - \frac{\sqrt{14}}{4} - \frac{1}{2}$

Decimal Form:

$x = 0.43541434 \dots, - 1.43541434 \dots$

Solve for x 4(4x+2)^2-19=37

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